\(ĐK:-\dfrac{1}{3}\le x\le2\\ PT\Leftrightarrow\left(\sqrt{3x+1}-2\right)-x+1-\sqrt{2-x}\left(\sqrt{2-x}-1\right)=0\\ \Leftrightarrow\dfrac{3\left(x-1\right)}{\sqrt{3x+1}+2}-\left(x-1\right)-\dfrac{\sqrt{2-x}\left(1-x\right)}{\sqrt{2-x}+1}=0\\ \Leftrightarrow\left(x-1\right)\left(\dfrac{3}{\sqrt{3x+1}+2}+\dfrac{\sqrt{2-x}}{\sqrt{2-x}+1}-1\right)=0\\ \Leftrightarrow\left[{}\begin{matrix}x=1\left(tm\right)\\\dfrac{3}{\sqrt{3x+1}+2}+\dfrac{\sqrt{2-x}}{\sqrt{2-x}+1}-1=0\end{matrix}\right.\)
Với \(x\ge-\dfrac{1}{3}\) thì \(\dfrac{3}{\sqrt{3x+1}+2}+\dfrac{\sqrt{2-x}}{\sqrt{2-x}+1}-1>0\)
Vậy pt có nghiệm duy nhất \(x=1\)
ĐKXĐ: \(-\dfrac{1}{3}\le x\le2\)
\(\sqrt{3x+1}=3-\sqrt{2-x}\) (do \(-\dfrac{1}{3}\le x\le2\Rightarrow3-\sqrt{2-x}\ge3-\sqrt{2+\dfrac{1}{3}}>0\))
\(\Leftrightarrow3x+1=9+2-x-6\sqrt{3-x}\)
\(\Leftrightarrow3\sqrt{2-x}=5-2x\)
\(\Leftrightarrow9\left(2-x\right)=\left(5-2x\right)^2\)
\(\Leftrightarrow4x^2-11x+7=0\)
\(\Leftrightarrow\left[{}\begin{matrix}x=1\\x=\dfrac{7}{4}\end{matrix}\right.\) (thỏa mãn)
